**3. Results and Discussion**

The analytical solution of Equation (17) is benchmarked with a finite element method (FEM) solution obtained by Ikegawa [9], whose dimensions of the liquid container are *h* = 0.6 m and *L* = 0.9 m. The vessel is exposed to the forced horizontal motion as given by *X* = 0.002 sin(5.5*t*). Figure 2 presents the time history of η (*x* = + *L*, t). The numerical result is denoted by circles and the analytical solution is denoted by a solid line. As shown, there is a good agreemen<sup>t</sup> between the analytical solution and numerical result with FEM.

An inspection of the analytical solution of Equation (22) is performed in Figure 3. The maximum distribution of the dimensionless entropy generation rate through the volume, with respect to time, is a

plot for various aspect ratios in that figure, according to (*Sg* = (ω2−ω2*i*)<sup>2</sup>*TSg* <sup>16</sup>πμ*<sup>X</sup>*2maxω<sup>4</sup> ). The axes in Figure 3 are Cartesian coordinate system of the vessel in accordance with Figure 1. The dimensions of rectangular storage tanks for each aspect ratio are *L* = (0.54α) 0.5 and *h* = (0.54/α) 0.5. As shown by the increase of the aspect ratio, the amount of entropy generated through the volume decreased. Further, the position of maximum entropy changed from the free surface to the side walls. It was expected that by an increase of the aspect ratio, the length of the tanks would increase and the dimensionless penetration length ( + υ*L*2ω ) would decrease. An increase of the aspect ratio made the dampening effects of sidewalls and bottom dominant in comparison with the free surface effects [10].

Horizontal periodic sway motions as *X* = Am sin (ω t) were applied to the rectangular storage tanks with different aspect ratios, namely the ratios of height to length of the rectangular storage tank (AR). Then, the effect of Am and ω was studied on the results. The oscillations of total entropy generation rate in the volume for the aspect ratio of α = 2.05 are plotted in Figure 4a. Similar to the time history of the wave, the entropy generation rate reaches its maximum after 10 periods of motion and decreases. The beating behavior of the entropy profile repeats as times goes on. To demonstrate the capability and accuracy of the present method, the results of the generated waves are compared with the available numerical calculations. Figure 4b takes from the results of [1]. Results from [1] are opposed to those stemming from this study, where Figure 4a should be compared with AR = 2.05 Figure 4b. The true unit for *Sg* obtained by the surface integration of volumetric entropy generation is (W/Km). However, in reference [1], as they considered the two-dimensional case witha1m depth, the unit appeared as (W/K). Moreover, the results show that an increase in the AR causes a decrease in the total entropy generation rate in the volume.

**Figure 4.** Entropy generation versus time for α = 2.05 (**a**) current study, (**b**) Reference [1].

Figure 4b was obtained by using the Reynolds-Averaged Navier–Stokes (RANS) and the Volume-of-Fluid (VOF) methods, together, in a commercial software solver [1]. The RANS equations were discretized and solved using the staggered grid finite difference and simplified marker and cell (SMAC) methods, and the available data were used for the model validation. By comparing the case of α = 2.05, it is clear that the trend and the order of magnitude of a maximum of entropy (2 × <sup>10</sup>−2) are the same. Since the current analytical solution is a suitable measure to decide on optimization based on the entropy generation, the entropy generation distribution offers designers with valuable data about the reasons for the energy loss.

Finally, Figure 5 reveals the value of the total entropy generation rate versus aspect ratio for α = 3. As shown, the trend of maximum entropy generation versus aspect ratio decreasing expects a peak point, which is caused by approaching the natural frequency of the system to the external forced frequency. Such phenomena lead to a local minimum point before the resonance, since the α = 1.4–1.5 is a candidate for the entropy minimization point. Generally, the overall function has no optimum and a higher aspect ratio leads to lower values of entropy generation.

**Figure 5.** Total entropy generation rate versus aspect ratio for α = 3.

As shown in reference [7] (see Figure 3c), 80 percent of energy of the fluid could be dissipated for the dimensionless frequencies in the range 0.95–1.05 (*f'* = *f*/*fn*), since the engineers try to design the sloshing vessels with the frequencies near to the structure frequency for highest energy absorbance rate. If the value of the energy of the fluid is symbolized by

$$E\_f = \frac{1}{2} m\_f (2\xi\_f \omega) X\_{\text{max}}^2 \omega^2 \tag{23}$$

and the work of no-conservative damping of the coupling structure are

$$E\_s = \frac{1}{2} m\_s (2\xi\_s \omega \nu) X\_{\text{max}}^2 \omega^2 \tag{24}$$

then the ratio of structure energy loss to the fluid loss is

$$
\gamma = \frac{\xi\_s m\_s}{\xi\_f m\_f}.\tag{25}
$$

The damping of the fluid could be estimated by the inverse of square root of the Galileo number (ratio of gravity forces and viscous forces)

$$\xi\_f = \frac{\upsilon^{1/2}}{L^{3/4} \mathfrak{g}^{1/4}} \tag{26}$$

where υ is the kinematic viscosity. The damping factor of 1–2% is predicted for fluids [10] (logarithmic decrement ≈ 6ξ) and 0.32% for solids [7], since the ratio of structural energy loss to fluid loss is approximated by 0.1–1 of the ratio of structural mass to fluid mass. As an example for engineering applications, the mass ratio of the tuned liquid damper to the solid structure is 1.05% [7], and then the amount of structure energy loss to the fluid loss is about 10–100. Subsequently, most of the energy dissipated in the solid part, which is not considered for optimization.

Since, as stated in the introduction section, for the specific case of sloshing, as such systems are used for the damping of the solid motion entropy of only fluid, they could not be a true objective function, and the energy dissipation in the structure should be considered, too. Today's practical meaning of EGM is very low. Although today engineers in the field of large vessels are mostly focused on frequency response design and exergy e fficiency is not considered in engineering code, the entropy minimization method is a growing topic in literature. In the current study, fluid entropy generation used as a measure of optimization of the sloshing phenomenon that is classified among free surface flows. The current research proposes future studies performing experiments for coupled cases with the sum energy dissipation of fluid and structure as an objective function.
